Plasma and Fusion Research: Regular Articles

Volume 9, 3402093 (2014)

Analytical Solution of High βp Equilibria with Natural Inboard
Poloidal Null Conﬁguration Obtained in the Spherical Tokamak
QUEST∗)
Kishore MISHRA1) , Hideki ZUSHI2) , Hiroshi IDEI2) , Saya TASHIMA1) , Santanu BANERJEE1) ,Makoto HASEGAWA2) , Kazuaki HANADA2) , Kazuo NAKAMURA2) , Akihide FUJISAWA2) ,Yoshihiko NAGASHIMA2) , Keisuke MATSUOKA2) , Arseny KUZMIN2) , Takumi ONCHI2)
and QUEST Team1)

IGSES, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
RIAM, Kyushu University, Kasuga, Fukuoka 816-8580, Japan

2)(Received 10 December 2013 / Accepted 22 April 2014)
High βp (εβp ∼ 1) equilibria obtained in a ECW heated Ohmic plasma is investigated using a simple analytic
solution of Grad-Shafranov equation. The formation of a natural inboard poloidal null associated with high βp
is explained consistently by high diamagnetism and negative triangularity. As βp is increased, the poloidal nullpoint penetrates further into the vacuum vessel, which is qualitatively explained by the analytic model. Transition
from inboard (high ﬁeld side) limiter bound to the natural divertor conﬁguration is associated with a reductionof the edge safety factor without appreciable enhancement of MHD activities. Such a scenario is also addressed
successfully with the model.
c 2014 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: tokamak plasma equilibrium, Grad-Shafranov equation, high beta plasma, natural divertor
DOI: 10.1585/pfr.9.3402093

1. Introduction
Spherical Tokamak is an attractive choice for future
fusion reactors for its ability to operate at high poloidal
beta (βp ) value, thereby increasing the bootstrap current
fraction. However, the maximum achievable βp is limited
by a so called equilibrium limit, where an Inboard Poloidal
ﬁeld Null (IPN) appears at the high ﬁeld side of the vacuum vessel. To describe such high βp equilibria and to
study its stability properties, analytical solution to GradShafranov equation (GSE) is often useful. Earlier work of
Solov’ev [1] with a simple pressure and poloidal current
proﬁle is extended [2–4] by including more linear stream
functions to describe high βp conﬁguration. The issue of
βp equilibrium limit has been addressed theoretically earlier [1–4]. Nevertheless, owing to the dearth of experimental demonstration of IPN conﬁguration, comparisonof such models with experimental data is far from adequate. The natural IPN conﬁguration is ﬁrst observed transiently in CDX-U and DIII-D during the current ramp-up
and closed ﬂux formation [5] phase, but no measurements
are done. In TFTR [6], a null point appeared at the inboard side in high βp plasma formed by ramping down theplasma current (Ip ) in a NBI heated plasma. Such highβp plasma production and sustainment required > 16 MW
of NBI power and could not be achieved at higher Ip .
author’s e-mail: mishra@triam.kyushu-u.ac.jp∗) This article is based on the presentation at the 23rd International Toki
Conference (ITC23).

IPN conﬁguration has not been realized or sustained effectively in tokamaks to date, albeit adequate demonstration of high βp plasma [7–9]. In this work, we present
the experimental IPN conﬁguration obtained rather easily
and ﬂexibly by creating an anisotropic pressure distribution of energetic electrons generated by injecting a modest
power (< 100 kW) of Electron Cyclotron Waves (ECW) in
the Ohmic plasma. Such high βp (∼ 4) plasma with IPN
conﬁguration is shown to be stable and various equilibrium and stability studies are performed using a simple
analytic solution of GSE. The equilibrium ﬂux functions
obtained from the model are compared with the results obtained from the magnetic measurements. With the help of
the model, plasma equilibrium is investigated as a function of plasma diamagnetism and triangularity and is compared with the experimental results for both the standard
limiter and IPN conﬁgurations. Important MHD surfacefunctions like safety factor and magnetic shear along with
βp are computed from the model and compared with themeasurements.
Outline of the paper is as follows: experimental detailof the IPN plasma formation is discussed next. Analyticalsolution of GSE is formulated in section 3. Section 4 provides an account of the computation of βp from the analytic
model and the variation of βp with plasma triangularity andthe null position. Finally, some conclusions are drawn insection 5.

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c 2014 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

Volume 9, 3402093 (2014)

2. High βp IPN Plasma Formation in
QUEST
QUEST is a medium sized spherical tokamak [10]with the major and minor radii of 0.68 and 0.4 m, respectively. The center stack (CS), which holds the Ohmic (OH)coil has an outer diameter of 0.2 m and the outer wall of
the vacuum vessel is at 1.4 m with the ﬂat divertor plates
at z ∼ ±1 m from the mid-plane. Inboard plasma boundary
is deﬁned by a limiter on the CS at 0.23 m. In the present
experiment, Ip is initiated by injecting ∼ 100 kW of ECW
power at 8.2 GHz for 150 ms from the two antennae locatedat the Low Field Side (LFS) followed by the OH phase. In
the OH phase, Ip is fed back to the OH coil power supply
in order to maintain it at −30 kA ± 10%. Here the negative
Ip is deﬁned clockwise as seen from the top of the torus.
The toroidal ﬁeld (Bt ) is set at 0.29 T at the fundamental
resonance Rfce = 0.33 m, where a vertical magnetic ﬁeld
Bz ∼ 25 mT is applied. At 2 s of the Ip ﬂattop, another
ECW pulse is injected. This results an increase in Ip from−30 kA to −32 kA in 0.3 s (Fig. 1 (a)). Due to the feedback
circuit, OH coil current is reversed transiently to produce
a retarding electric ﬁeld (positive loop voltage VL ) in order to bring back Ip to the feedback value. Plasma shapeand position are identiﬁed by the magnetic measurementsfrom the 64 ﬂux loops outside the plasma. The magnetic
ﬂux contours are reconstructed by a ﬁtting procedure described in ref. 11 and are shown in Fig. 1 (b). It can beseen that plasma conﬁguration is changed from an inboard
limiter (IL) during OH phase to IPN in the ECW phase.
A tangentially viewing visible camera image also conﬁrms

such transition in conﬁguration, where the separatrix strike
points on the limiter are clearly seen. The poloidal magnetic null point (Rin ) measured by the ﬂux loops is shownin Fig. 1 (a), which distinctly shows that it has moved inside the vacuum vessel up to 0.4 m in the ECW phase. The
natural appearance of such inboard null is a consequenceof high βp formation. We evaluate βp from the generalized
Shafranov equation for radial force balance for the equilibrium vertical magnetic ﬁeld [12]
8R0Bθ (a)
li 3
ln
(1)
Bz =
+ βp + − ,2a2 2
where Bθ (a) = μ0 Ip /2πa is the poloidal magnetic ﬁeld at
the boundary r = a. li is the internal inductance assumed
here to be unity and not varying.
The IPN plasma has an aspect ratio A = R0 /a ∼ 1.7
to 2.6, hence equation (1) is deemed valid to describe thepresent equilibria in particular. βp evaluated in this method
shows an increment from 2.4 in OH phase to ∼ 4 in the
ECW phase in < 0.18 s. In order to maintain this high βp
plasma in equilibrium, Bz is increased from 26 mT in the
OH phase to 33 mT during the ECW phase, or else Ip is
not sustained. Requirement of high Bz is also evident fromthe fact that, the increased pressure gradient (∇p = j × B)
needs to be balanced by enhancing B at a ﬁxed Ip .
It can be noted here that such high βp plasma is stablefrom 2.5 - 3.5 s and there are no observable MHD instabilities. The kink safety factor q∗ = πεa(1 + κ2 )B0 /μ0 Ip fromthe measurements shows that it is reduced from 4.1 (limiter) to 1.6 in the IPN conﬁguration (Fig. 1 (a)). This is dueto the reduction of plasma minor radius a, larger R0 andreduced B0 due to the shift of the plasma outward.

3. Analytic Solution of IPN Equilibria
For toroidal axisymmetric equilibria (independent of
toroidal angle ϕ) magnetic ﬁeld can be represented by B =(1/R)∇ψ×eϕ + Bϕ ∇ϕ , where ψ is the poloidal magnetic ﬂux
function, a constant on nested magnetic surfaces, ϕ is thetoroidal angle of (R, Z, ϕ) cylindrical co-ordinate systemand eϕ is unit vector along ϕ. Deﬁning G(ψ) as the poloidal
current function so as to generate Bϕ = μ0G/2πR, GSE forsuch equilibrium can be expressed as,


∂p
∂G
∂ 1 ∂ψ ∂2 ψ+
= −μ20G− μ0 (2πR)2 . (2)R∂R R ∂R ∂Z 2
∂ψ∂ψFig. 1 Time traces of plasma current (Ip ), one turn loop voltage(VL ), kink safety factor (q∗ ). The transient spike in βp
at ∼2.3 s is due to a minor disruption in Ip . The inboard
plasma position (Rin ) depicts separatrix formation (Rin >0.22 m) as βp increased with ECW injection. Magnetic
ﬂux contours during OH phase (1.95 s) and ECW phase(2.18 s) show the formation of IPN conﬁguration.

GSE, in general needs to be solved numerically becauseof its non-linearity. However, a number of powerful yet
simpler models to solve GSE analytically are described inrefs. [1–3] and they have their inherent ability to choose
geometrical parameters independently. We have here used
the analytical solution discussed in ref. 2 for the sphericaltorus plasma equilibrium in particular, given as

c0 σ2
Z2 2
2
2 2
(R
ψ(R, Z) =−R)+
4(R − ωR2a )a
8 (1+ σ2 )
σ2


R2
2
2
2
2
(3)
−2τRa R ln 2 − R + Ra ,Ra

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where c0 is related to pressure gradient, Ra represents magnetic axis and ω, σ are functions of plasma geometricalparameters through some complex functions of plasma diamagnetism factor τ. The factor τ is particularly importantin spherical torus high beta plasma equilibria. For τ set tozero, equation (3) is reduced to Solov’ev plasma equilibrium solution. Although plasma diamagnetism and plasma
shape, especially plasma triangularity, are coupled witheach other, but the solution given above allows to choose
the two parameters independently. Deﬁning the plasmaboundary value of poloidal ﬂux as ψb = (c0 /8)[σ2 /(1 +σ2 )]R4b , where Rb is a constant, the normalized poloidal
magnetic ﬂux function can be deﬁned as,
Z2ψbψR, Z = 4 (R2 − R2a )2 + 4 2 (R2 − ωR2a )
σ
Rb


R2
(4)
−2τR2a R2 ln 2 − R2 + R2a .
Ra

Fig. 2 Boundary ﬂux contours for IL and IPN plasma are compared from the analytic model and the ﬂux loop reconstruction. The model parameters for both the cases are
inscribed.

The constants in the above equation can be expressed asfunctions of plasma geometrical parameters like, inverse
aspect ratio ε, elongation k, triangularity δ [2].
Ra
= f (ε, τ, R0 ), ω = g(τ, ε, σ, Ra , Rb ),Rb
σ = h(κ, ε, σ, τ, Ra ).

(5)

Here τ and c0 are chosen iteratively such that the imposedboundary condition is satisﬁed while constraining the parallel surface current λ(ψ) = μ0 (GI  − IG )/(qG − I), atψ = ψb to zero. Here, both I (toroidal current) and q (safety
factor proﬁle) are ﬂux functions and the prime denotes differentiation with respect to the ﬂux function. All thesefunctions are obtained from the two basic surface functions U(ψ) and V(ψ) evaluated by numerical integration as
described in ref. [2].
In ref. 2, no constraint on Ip is imposed and it is calculated self consistently. However in the present case wehave forced the total Ip bounded by ψ = ψb to be ﬁxed
and is set to the measured Ip in addition to the λ(ψb ) = 0
boundary condition. By doing so we could choose the constant c0 corresponding to the actual Ip . This simpliﬁed theiteration procedure and one only needed to ﬁnd the optimum τ so as to satisfy the boundary condition. The plasma
geometrical parameters ε, k, δ, R0 inferred from the magnetic measurements, Ip measured from the Rogowsky coiland the applied toroidal magnetic ﬁeld B0 are given as the
input parameters to the model. ψ(R, Z) are the output of
the model, which are used to ﬁnd boundary ﬂux contour
as deﬁned by the inner limiter. Such boundary ﬂux contours computed from the model are superimposed on the
LCFS inferred from the magnetic measurements, and areshown in Fig. 2. It is seen that the analytic solution agrees
excellently with the ﬂux loop reconstruction. It is also important to correctly choose τ as it has a direct bearing on
the resulting βp and plasma boundary. For a critical valueof τ, a ﬁeld null just appears inside the vessel, which is thelimiting βp for IL to IPN transition as discussed in the next

Fig. 3 Magnetic shear (S ) and safety factor (q) as ﬂux functions
computed from the analytic model: left is limiter plasma;right is for IPN plasma.

section. However, the equilibria shown below at 1.95 s isperfectly in IL conﬁguration.
Once a convergence of the solution is found at theboundary, it is easy to ﬁnd various ﬂux functions like magnetic well, magnetic shear S (ψ), safety factor q(ψ) etc. In
Fig. 3, S and q for the two conﬁgurations (IL and IPN)
are shown. From q(ψ) proﬁle it is seen that q95 is reduced from 4.3 in IL to 1.5 in the IPN conﬁguration andsuch reduction of q95 during the IPN formation is consistent with the measured q∗ as shown in the Fig. 1. The
central q (see Fig. 3) for the IPN plasma is marginally below 1. Absence of any conﬁrmed sawteeth oscillationsin these discharges, however, suggests that central q may
not be below 1. Considering ECW heating in low density
plasma (ne_ave < 8 × 1017 m−3 ), strong pressure anisotropy(p⊥  p) is expected. The analytic solution presented here
however, assumes an isotropic pressure proﬁle. It has beenshown earlier [13] that the estimation of central q can besigniﬁcantly underestimated by this assumption. The roleof pressure anisotropy in the present equilibrium is left asa future work.

4. Poloidal Beta
Computation of βp from the analytic model is quite
intriguing, particularly in IPN plasma. The hunting for op-

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Fig. 4 Poloidal beta (βp ) computed from the analytic solution(circles) and magnetic measurements (cross) as a function of null position (Rnull ) and triangularity. D is the triangularity parameter in the analytic model and δ is themeasured plasma triangularity. The values of model parameters for diﬀerent points are chosen to best match the
LCFS, E = 0.85 - 1.43, D = 0 to −0.56, H = Hc .

timum τ corresponding to λ(ψb ) minimum is often cumbersome in the method described in the previous section.
Additionally from Fig. 1 it is seen that as βp increases beyond 2.5 with application of ECW, Rnull (= Rin ) moves into the vacuum vessel and has a very good correlation with
βp . So to study the IPN plasma and relation of βp with Rnull
more easily, we dwelt on the treatment described in theref. 3, where IPN equilibria can be studied more ﬂexibly
by introducing the following transformations
ωσand H = −2τ,(6)D=
, E= √1−ω
1−ω
where, D represents plasma triangularity, E relates toplasma elongation and H is related to diamagnetism. The
analytical GSE in this case is same as eq. 4. As H is increased, a poloidal ﬁeld null appeared at the high ﬁeld side.
Correspondingly the critical H is deﬁned as a function of
Rnull as Hc = 2(1 − R2null )/ ln R2null , where the null point justabout to appear. This introduces much ﬂexibility in computing boundary ﬂux functions including the poloidal magnetic ﬁeld. βp = 2μ0 P v / Bp 2 is calculated as describedin ref. 3. In Fig. 4 we have computed βp as a function of
Rnull and D from the analytic model for various IPN equilibria corresponding to H = Hc to reproduce high βp conﬁgurations. βp shows an increasing trend both with larger
Rnull position and higher negative values of D for the rangeof E = 0.85 - 1.43. This prediction is consistent with themeasured values of βp as shown in the Fig. 4 at discrete
time intervals. Figure 4 (a) shows a critical βcp ∼ 2.5 atwhich the IL-IPN transition occurs and is supported by themodel. In Fig. 5 we show the variation of βp as a functionof E and H for diﬀerent D values. It shows that βp canbe increased with higher elongation (E) and larger negative triangularity. It also shows that for higher diamagnetic
factor (H), βp is increased. Plasma boundary is largely decided by the choice of these parameters, however, in allthese cases, at a critical Hc or higher negative triangularity, IPN conﬁguration is achieved for βcp >∼ 2.5. In somecases, IPN conﬁguration is also achieved at a marginallylower βp < βcp if H > Hc . However, such a scenario isFig. 5 Poloidal beta (βp ) computed from the analytic solution
as a function of E and H for diﬀerent D parameters at
ﬁxed Rnull = 0.3 m. Higher elongation, diamagnetism andnegative triangularity, all are favorable for increasing βp .

not obtained in the present experiment. The boundary solutions obtained in the analytic model satisfying the highβp conﬁguration (Fig. 4) are found to be for κ ∼ 1, which
is higher than the ﬁtted boundary obtained from the magnetic measurements (0.6 < κ < 0.7). This diﬀerence can be
attributed to the consideration of simple isotropic pressure
proﬁle or the choice of current proﬁle, which requires further study. However, the simple analytic solution can qualitatively explain the equilibrium characteristics of high βp
IPN plasma conﬁgurations obtained in our experiment.

5. Conclusion
High βp equilibrium is obtained in a ECW heated OH
target plasma in QUEST. With high βp , plasma conﬁguration is transformed from an inboard limiter to inboard
natural divertor with a poloidal ﬁeld null appearing at the
inboard side. A simple analytic solution is demonstrated
to explain this phenomenon, which agrees well with themeasurements. This further deﬁnes a critical βp for thetransition from IL to IPN. The model also shows that βp
increases with a higher value of negative triangularity, consistent with the measurements and predicts that βp can beraised by suitable shaping of plasma to have a negative triangularity. The reduced q with IPN formation is also consistent with the analytic model prediction. However theapparent low value of central q indicated by the present
model may need to be revisited by including anisotropicpressure in the model.

Acknowledgement

This work is supported by Grant-in-aid for ScientiﬁcResearch (S24226020, A21246139). This work is also
performed with the support and under the auspices of the
NIFS Collaboration Research Program (NIFS11KUTR06
3).
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